On the intersections of circuits and cocircuits in matroids (Q762168)

A 3- or 4-element set is called a triad or a quad, respectively, if it is the intersection of a circuit and a cocircuit of a matroid. \textit{P. D. Seymour} [Combinatorica 1, 387-394 (1981; Zbl 0489.05020)] proved that a matroid has a triad if and only if it is non-binary; and then every pair of elements is contained in a triad. The author characterizes those matroids which have a quad. He also shows that if a matroid has a circuit and a cocircuit meeting in more than 4 elements then it has a quad as well. Finally he proves that if a matroid has a quad and is 3-connected then every pair is in a quad.